DOCSLIB.ORG

The Three Laws of Thought, Plus One: the Law of Comparisons

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Three Laws of Thought, Plus One: the Law of Comparisons Axioms 2014, 3, 46-49; doi:10.3390/axioms3010046 OPEN ACCESS axioms ISSN 2075-1680 www.mdpi.com/journal/axioms/ Concept Paper The Three Laws of Thought, Plus One: The Law of Comparisons Thomas L. Saaty Joseph M. Katz Graduate School of Business, University of Pittsburgh; Pittsburgh, PA 15260, USA; E-Mail: [email protected] Received: 30 December 2013; in revised form: 21 January 2014 / Accepted: 21 January 2014 / Published: 10 February 2014 Abstract: The rules of logic are nearly 2500 years old and date back to Plato and Aristotle who set down the three laws of thought: identity, non-contradiction, and excluded middle. The use of language and logic has been adequate for us to develop mathematics, prove theorems, and create scientific knowledge. However, the laws of thought are incomplete. We need to extend our logical system by adding to the very old laws of thought an essential yet poorly understood law. It is a necessary law of thought that resides in our biology even deeper than the other three laws. It is related to the rudiments of how we as living beings, and even nonliving things, respond to influences as stimuli. It helps us discriminate between being ourselves and sensing that there is something else that is not ourselves that even amoebas seem to know. It is the intrinsic ability to sense and distinguish. This fourth law is the law of comparisons. Although it has been missing from our logical deductions it underlies the other three laws of thought because without it we cannot know what is and what is not. Keywords: laws of thought; new law of comparisons; Plato and Aristotle’s axioms; Arthur Schopenhauer The original three laws of thought, defined by Plato (Figure 1), comprise the most basic requirements for identification and reasoning: First, that nothing can become greater or less, either in number or magnitude, while remaining equal to itself. Second, that without addition or subtraction there is no increase or diminution of anything, but only equality. Third, that what was not before cannot be afterwards, without becoming and having become. Axioms 2014, 3 47 Figure 1. Plato (Left) Holding the Timaeus and Aristotle Holding the Ethics, a Painting by Raphael Sanzio at the School of Athens. We turn now to the modern way of introducing these three laws: The Three Laws, Plus One Law 1: The Law of Identity Two entities x and y are identical if they share the same properties. A necessary and sufficient condition for two entities x and y to be identical (be the same) is that they share the same properties: ∀x ∀ y[ x = y →∀ P (] Px ↔ Py and ∀∀x y[( ∀ P Px ↔ Py ) → x = y ]. In other words: for all x and all y, the statement x is y implies that for every property P, if x has that property, then y has that property and conversely. It is also true that for all properties, if x and y have the same properties, then they are identical. If x is not identical to y for all y, then it is identical with x, enabling one to make the distinction of x from all y ≠ x. According to Leibnitz, the law of identity is the first primitive truth of reason that is affirmative. Yet, one can discern from the statement of this law that identity in itself has no meaning unless one thinks of it in terms relative to other things. Thus satisfying the law of identity involves the even more basic implicit assumption that entities must be compared with all other entities to know if they are identical with themselves or there are other entities identical with them. Because it is not possible to know if an entity is identical with itself unless it is previously known from memory, it is necessary that it should be compared with itself. Comparisons are necessary and sufficient to satisfy the law of identity. One may conclude that to know something is identical only with itself, we must compare it with all other things for all the properties it has. After numerous comparison of x with other y, we learn to characterize x by remembering it according to the properties it has and their intensities. We also need to compare property P with every other property Q as to which is possessed by x and which is not. Axioms 2014, 3 48 Law 2: The Law of Noncontradiction Either x or not x is true and there is no other possibility. This means that opposite assertions cannot be true at the same time. Here again, two entities are involved, x and not x, and to determine which it is, distinction between them needs to be made through comparison. In logic, to show that x holds is equivalent to showing that not x does not hold. Law 3: The Law of the Excluded Middle An entity x has or does not have a property P. Conversely, a property P is either possessed by an entity x or it is not possessed by x. This law does not allow for different degrees of the property P. To talk about degrees of properties is clearly different from the black and white distinctions that Laws 2 and 3 require. Further, to identify x, as in Law 1, we have to know not only what properties it has, but also to what extent it has these properties. Thus it appears that knowledge about comparisons must precede these three laws, and we must introduce a new law that precedes them. Once we admit that comparisons are a biological talent and that we need to be more precise about understanding things, we can elaborate deterministic outcomes into all shades of gray, which in fact we human beings are able to detect. Law 4: The Law of Quantitative Comparison [1,2,3,4] Given two entities x and y both having a property P. Either x has P more than y, less than y or is indifferent to y. When x and y are closely similar (homogeneously close) it is possible be using expert judgment to estimate the dominance of one over another numerically and derive priorities from such comparisons. Let be a finite set of n entities. Let ℑ be a set of properties or attributes with respect to which entities in are compared by estimating the similarity between one entity and another. A property is a feature that an object or individual possesses even if we are ignorant of this fact, whereas an attribute is a special concept assigned to some entity under consideration. Sometimes erroneously an attribute is referred to as a property. When two entities in are compared according to a property P in ℑ, we say that we are performing ℑ binary comparisons. Let >P be a binary relation representing “more preferred than” or “dominates” ℑ with respect to a property P in . Let ~P be the binary relation “indifferent to” with respect to a ℑ ℑ > > property P in . Hence, given two entities, Ai , Aj in , either AAijP or AAjiP or AAij~P for all P in ℑ. We use to indicate more preferred or indifferent. A given family of binary relations ℑ >P with respect to a property P in is a primitive concept. We have used this kind of relation to derive the notion of priority or importance both with respect to one or several properties. Surprisingly we found a similarity between the foregoing axioms and those given by Arthur Schopenhauer [4] who was not equipped to develop a mathematical theory to use them. He listed them in the following way in his On the Fourfold Root of the Principle of Sufficient Reason, §33: Axioms 2014, 3 49 a. A subject is equal to the sum of its predicates, or a = a. b. No predicate can be simultaneously attributed and denied to a subject, or a ≠ ~a. c. Of every two contradictorily opposite predicates, one must belong to every subject. d. Truth is the reference of a judgment to something outside it as its sufficient reason or ground. Because the brain works with electricity, that is quantitative in nature, we need to talk about the use of mathematics that derives from making comparisons. This mathematics cannot be intrinsically based on the use of Cartesian axes to represent independent and dependent variables measured with arbitrary units applied uniformly, particularly because properties do not arise in our understanding as space and time that are endowed with homogeneity throughout. Had the mathematics of relative measurement been known long ago, our understanding of the world would (not “might”) have been different because it deals with interdependence in a natural way. Later, for example, we shall see how the inverse square laws of nature are an intrinsic part of the Fourier transform of our neural firings in the real number domain. Relative measurement applies to things that do not change as well as to things that do. If the world began with the big bang, everything was energy with dynamic behavior over time. Accurate measurement did not last long. When Descartes invented the ideas of axes and scales with arbitrary units, he assumed that the world was fixed and he could study it by using the axes. However, the world of human behavior changes, and situations change. Conflicts of Interest The authors declare no conflict of interest. References 1. Saaty, T.L. The Analytic Hierarchy Process; McGraw Hill: New York, NY, USA, 1980. 2. Saaty, T.L. Fundamentals of Decision Making; RWS Publications: Pittsburgh, PA, USA, 2006; pp. 1–478. 3. Saaty, T.L. Principia Mathematica Decernendi, Subtitle: Mathematical Principles of Decision Making; RWS Publications: Pittsburgh, PA, USA, 2009; pp.
Load more

Recommended publications
  • Contradiction Or Non-Contradiction?  Hegel’S Dialectic Between Brandom and Priest
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Padua@research CONTRADICTION OR NON-CONTRADICTION? HEGEL’S DIALECTIC BETWEEN BRANDOM AND PRIEST by Michela Bordignon Abstract. The aim of the paper is to analyse Brandom’s account of Hegel’s conception of determinate negation and the role this structure plays in the dialectical process with respect to the problem of contradiction. After having shown both the merits and the limits of Brandom’s account, I will refer to Priest’s dialetheistic approach to contradiction as an alternative contemporary perspective from which it is possible to capture essential features of Hegel’s notion of contradiction, and I will test the equation of Hegel’s dialectic with Priest dialetheism. 1. Introduction According to Horstmann, «Hegel thinks of his new logic as being in part incompatible with traditional logic»1. The strongest expression of this new conception of logic is the first thesis of the work Hegel wrote in 1801 in order to earn his teaching habilitation: «contradictio est regula veri, non contradictio falsi»2. Hegel seems to claim that contradictions are true. The Hegelian thesis of the truth of contradiction is highly problematic. This is shown by Popper’s critique based on the principle of ex falso quodlibet: «if a theory contains a contradiction, then it entails everything, and therefore, indeed, nothing […]. A theory which involves a contradiction is therefore entirely useless I thank Graham Wetherall for kindly correcting a previous English translation of this paper and for his suggestions and helpful remarks. Of course, all remaining errors are mine.
    [Show full text]
  • A New Logic for Uncertainty
    A New Logic for Uncertainty ∗ LUO Maokang and HE Wei† Institute of Mathematics Institute of Mathematics Sichuan University Nanjing Normal University Chengdu, 610064 Nanjing, 210046 P.R.China P.R.China Abstract Fuzziness and randomicity widespread exist in natural science, engineering, technology and social science. The purpose of this paper is to present a new logic - uncertain propositional logic which can deal with both fuzziness by taking truth value semantics and randomicity by taking probabilistic semantics or possibility semantics. As the first step for purpose of establishing a logic system which completely reflect the uncertainty of the objective world, this logic will lead to a set of logical foundations for uncertainty theory as what classical logic done in certain or definite situations or circumstances. Keywords: Fuzziness; randomicity; UL-algebra; uncertain propositional logic. Mathematics Subject Classifications(2000): 03B60. 1 Introduction As one of the most important and one of the most widely used concepts in the whole area of modern academic or technologic research, uncertainty has been involved into study and applications more than twenty years. Now along with the quickly expanding requirements of developments of science and technology, people are having to face and deal with more and more problems tangled with uncertainty in the fields of natural science, engineering or technology or even in social science. To these uncertain prob- arXiv:1506.03123v1 [math.LO] 9 Jun 2015 lems, many traditional theories and methods based on certain conditions or certain circumstances are not so effective and powerful as them in the past. So the importance of research on uncertainty is emerging more and more obviously and imminently.
    [Show full text]
  • Absolute Contradiction, Dialetheism, and Revenge
    THE REVIEW OF SYMBOLIC LOGIC,Page1of15 ABSOLUTE CONTRADICTION, DIALETHEISM, AND REVENGE FRANCESCO BERTO Department of Philosophy, University of Amsterdam and Northern Institute of Philosophy, University of Aberdeen Abstract. Is there a notion of contradiction—let us call it, for dramatic effect, “absolute”— making all contradictions, so understood, unacceptable also for dialetheists? It is argued in this paper that there is, and that spelling it out brings some theoretical benefits. First it gives us a foothold on undisputed ground in the methodologically difficult debate on dialetheism. Second, we can use it to express, without begging questions, the disagreement between dialetheists and their rivals on the nature of truth. Third, dialetheism has an operator allowing it, against the opinion of many critics, to rule things out and manifest disagreement: for unlike other proposed exclusion-expressing-devices (for instance, the entailment of triviality), the operator used to formulate the notion of absolute contradiction appears to be immune both from crippling expressive limitations and from revenge paradoxes—pending a rigorous nontriviality proof for a formal dialetheic theory including it. Nothing is, and nothing could be, literally both true and false. [. ] That may seem dogmatic. And it is: I am affirming the very thesis that [the dialetheists] have called into question and—contrary to the rules of debate—I decline to defend it. Further, I concede that it is indefensible against their challenge. They have called so much into question that I have no foothold on undisputed ground. So much the worse for the demand that philosophers always must be ready to defend their theses under the rules of debate.
    [Show full text]
  • Russell's Theory of Descriptions
    Russell’s theory of descriptions PHIL 83104 September 5, 2011 1. Denoting phrases and names ...........................................................................................1 2. Russell’s theory of denoting phrases ................................................................................3 2.1. Propositions and propositional functions 2.2. Indefinite descriptions 2.3. Definite descriptions 3. The three puzzles of ‘On denoting’ ..................................................................................7 3.1. The substitution of identicals 3.2. The law of the excluded middle 3.3. The problem of negative existentials 4. Objections to Russell’s theory .......................................................................................11 4.1. Incomplete definite descriptions 4.2. Referential uses of definite descriptions 4.3. Other uses of ‘the’: generics 4.4. The contrast between descriptions and names [The main reading I gave you was Russell’s 1919 paper, “Descriptions,” which is in some ways clearer than his classic exposition of the theory of descriptions, which was in his 1905 paper “On Denoting.” The latter is one of the optional readings on the web site, and I reference it below sometimes as well.] 1. DENOTING PHRASES AND NAMES Russell defines the class of denoting phrases as follows: “By ‘denoting phrase’ I mean a phrase such as any one of the following: a man, some man, any man, every man, all men, the present king of England, the centre of mass of the Solar System at the first instant of the twentieth century, the revolution of the earth around the sun, the revolution of the sun around the earth. Thus a phrase is denoting solely in virtue of its form.” (‘On Denoting’, 479) Russell’s aim in this article is to explain how expressions like this work — what they contribute to the meanings of sentences containing them.
    [Show full text]
  • Florensky and Frank
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE Po zn ań skie Studia Teologiczne Tom 22, 2008 ADAM DROZDEK Duquesne University, Pittsburgh, Pennsylvania Defying Rationality: Florensky and Frank Przezwyciężyć racjonalność: Fiorenski i Frank What should be the basis of knowledge in theological matters? The question has been pondered upon by theologians for centuries and not a few proposals were made. One of these was made by the Orthodox priest and scientist Pavel Florensky in The pillar and ground of the truth (1914) considered to be the most original and influen­ tial work of the Russian religious renaissance1, marking “the beginning of a new era in Russian theology”2 and even “one of the most significant accomplishments of ec­ clesiastical thinking in the twentieth century”3. Proposals made in a work deemed to be so important certainly arrest attention. I. THE LAW OF IDENTITY A centerpiece of rationalism, according to Florensky, is the law of identity, A = A, which is the source of powerlessness of rational reasoning. “The law A = A be­ comes a completely empty schema of self-affirmation” so that “I = I turns out to be ... a cry of naked egotism: ‘I!’ For where there is no difference, there can be no con- 1 R. Slesinski, Pavel Florensky: a metaphysics o f love, Crestwood: St. Vladimir’s Seminary Press 1984, 22. 2 N. Zernov, The Russian religious renaissance of the twentieth century, London: Darton, Longman & Todd 1963, 101. 3 M. Silberer, Die Trinitatsidee im Werk von Pavel A. Florenskij: Versuch einer systematischen Darstellung in Begegnung mit Thomas von Aquin, Wtirzburg: Augustinus-Verlag 1984, 254.
    [Show full text]
  • Overturning the Paradigm of Identity with Gilles Deleuze's Differential
    A Thesis entitled Difference Over Identity: Overturning the Paradigm of Identity With Gilles Deleuze’s Differential Ontology by Matthew G. Eckel Submitted to the Graduate Faculty as partial fulfillment of the requirements for the Master of Arts Degree in Philosophy Dr. Ammon Allred, Committee Chair Dr. Benjamin Grazzini, Committee Member Dr. Benjamin Pryor, Committee Member Dr. Patricia R. Komuniecki, Dean College of Graduate Studies The University of Toledo May 2014 An Abstract of Difference Over Identity: Overturning the Paradigm of Identity With Gilles Deleuze’s Differential Ontology by Matthew G. Eckel Submitted to the Graduate Faculty as partial fulfillment of the requirements for the Master of Arts Degree in Philosophy The University of Toledo May 2014 Taking Gilles Deleuze to be a philosopher who is most concerned with articulating a ‘philosophy of difference’, Deleuze’s thought represents a fundamental shift in the history of philosophy, a shift which asserts ontological difference as independent of any prior ontological identity, even going as far as suggesting that identity is only possible when grounded by difference. Deleuze reconstructs a ‘minor’ history of philosophy, mobilizing thinkers from Spinoza and Nietzsche to Duns Scotus and Bergson, in his attempt to assert that philosophy has always been, underneath its canonical manifestations, a project concerned with ontology, and that ontological difference deserves the kind of philosophical attention, and privilege, which ontological identity has been given since Aristotle.
    [Show full text]
  • Vasiliev and the Foundations of Logic
    Chapter 4 Vasiliev and the Foundations of Logic Otávio Bueno Abstract Nikolai Vasiliev offered a systematic approach to the development of a class of non-classical logics, which he called “Imaginary Logics”. In this paper, IexaminecriticallysomeofthecentralfeaturesofVasiliev’sapproachtological theory, suggesting its relevance to contemporary debates in the philosophy of logic. IarguethatthereismuchofsignificantvalueinVasiliev’swork,whichdeserves close philosophical engagement. Keywords Vasiliev • logical pluralism • Revisability • a priori • negation 4.1 Introduction: Six Central Features of Vasiliev’s Approach to Logical Theory Nikolai Vasiliev’s approach to logical theory has a number of features. Six of them, in particular, are worth highlighting: (a) logical pluralism (there is a plurality of logics, depending on the subject matter under consideration); (b) logical revisability (certain logical laws can be revised depending on the subject matter); (c) logical non-a priorism (certain logical laws are empirically based); (d) logical contingency (given the empirical nature of some logical laws, they are ultimately contingent; in this context, issues regarding the scope of logic are also examined, with the accompanying distinction between laws of objects and laws of thought); (e) the nature of negation (negation is characterized via incompatibility; it is not just difference, nor is it grounded on absence, and it is inferred rather than perceived); and (f) logical commitment (why Vasiliev is not a dialetheist, after all). In this paper, I will examine each of these features, and suggest the relevance of many of Vasiliev’s proposals to contemporary philosophical reflection about the foundations of logic. Although the terms I use to describe some of the views O. Bueno (!) Department of Philosophy, University of Miami, Coral Gables, FL 33124, USA e-mail: [email protected] ©SpringerInternationalPublishingAG2017 43 V.
    [Show full text]
  • The Other Side of Peirce's Phaneroscopy: Questioning And
    The Other Side of Peirce’s Phaneroscopy: Questioning and Analogising Phaneron... 1 without ‘Being’ Iraklis Ioannidis (University of Glasgow) Abstract Research on Peirce’s phaneroscopy has been done with and through the paradigm or the conceptual schema of “Being” — what has been cri- tiqued by post-structuralist philosophers as the metaphysics of Being. Thus, such research is either limited to attempts to define “phaneron,” or to identify whether there is a particular and consistent meaning intention behind Peirce’s use of this term. Another problematic characteristic with such a way of engaging with phaneroscopy is the very anonymity of the schema of “Being.” While all scholars admit to the universality of “phaneron,” rarely, if ever, do we see an account of how such universal- ity can be instantiated. In this paper, I attempt to engage with phan- eroscopy differently. Instead of presenting a better version of what phaneroscopy is, or making arguments about what is the case with phan- eroscopy, both of which are ways of philosophising with “being,” I at- tempt to enact phaneroscopy. This would mean to undertake to follow Peirce’s instructions for the phaneroscopist and report the findings. Based on the latter, I shall analogise phaneron with the possibility of understanding. Finally, instead of having a conclusion which would im- ply an intention of making a case, and thus closure, I shall open up the 1 The first version of this paper was presented at the “Pragmatism and the Ana- lytic — Continental Split” conference held at the University of Sheffield in Au- gust 2017. I would like to thank Professor Shannon Dea, Professor James Wil- liams, and Dr.
    [Show full text]
  • Abstract Consequence and Logics
    Abstract Consequence and Logics Essays in honor of Edelcio G. de Souza edited by Alexandre Costa-Leite Contents Introduction Alexandre Costa-Leite On Edelcio G. de Souza PART 1 Abstraction, unity and logic 3 Jean-Yves Beziau Logical structures from a model-theoretical viewpoint 17 Gerhard Schurz Universal translatability: optimality-based justification of (not necessarily) classical logic 37 Roderick Batchelor Abstract logic with vocables 67 Juliano Maranh~ao An abstract definition of normative system 79 Newton C. A. da Costa and Decio Krause Suppes predicate for classes of structures and the notion of transportability 99 Patr´ıciaDel Nero Velasco On a reconstruction of the valuation concept PART 2 Categories, logics and arithmetic 115 Vladimir L. Vasyukov Internal logic of the H − B topos 135 Marcelo E. Coniglio On categorial combination of logics 173 Walter Carnielli and David Fuenmayor Godel's¨ incompleteness theorems from a paraconsistent perspective 199 Edgar L. B. Almeida and Rodrigo A. Freire On existence in arithmetic PART 3 Non-classical inferences 221 Arnon Avron A note on semi-implication with negation 227 Diana Costa and Manuel A. Martins A roadmap of paraconsistent hybrid logics 243 H´erculesde Araujo Feitosa, Angela Pereira Rodrigues Moreira and Marcelo Reicher Soares A relational model for the logic of deduction 251 Andrew Schumann From pragmatic truths to emotional truths 263 Hilan Bensusan and Gregory Carneiro Paraconsistentization through antimonotonicity: towards a logic of supplement PART 4 Philosophy and history of logic 277 Diogo H. B. Dias Hans Hahn and the foundations of mathematics 289 Cassiano Terra Rodrigues A first survey of Charles S.
    [Show full text]
  • The Development of Laws of Formal Logic of Aristotle
    This paper is dedicated to my unforgettable friend Boris Isaevich Lamdon. The Development of Laws of Formal Logic of Aristotle The essence of formal logic The aim of every science is to discover the laws that could explain one or another phenomenon. Once these laws are discovered, then science proceed to study the other phenomena, which in the nature are of an infinite set. It is interesting to note that in the process of discovering a law, for example in physics, people make thousands of experiments, build proves, among them some experience, or evidence - useful for understanding a certain phenomenon, but other experiments or evidence proved fruitless. But it is found only with hindsight, when the law is already discovered. Therefore, with the discovering of the law it is enough to show 2 - 3 experiments or prove to verify its correctness. All other experiments were the ways of study and there is no need to repeat them, to understand how the law works. In the exact sciences, it is understood, and therefore the students studies only the information that is necessary to understand specific phenomena. By no means this is the case with the study of formal logic. Formal logic, as opposed to other sciences: physics, chemistry, mathematics, biology and so on, studies not an infinite number of phenomena in nature, but only one how a man thinks, how he learns the world surrounding us, and how people understand each other. In other words, what laws govern the logic of our thinking, i.e., our reasoning and judgments in any science or in everyday life.
    [Show full text]
  • In Defense of the Law of Noncontradiction Imply Is Both True and False
    Edward N. Zalta 2 aletheists argue that some sentences are both true and false, and that sometimes appearances are not deceiving—there just are special groups of true yet jointly incompatible sentences for which the contradiction they In Defense of the Law of Noncontradiction imply is both true and false. We shall suppose, for the purposes of this paper, that the law of noncontradiction is the claim that there are no ∗ true contradictions. Thus, dialetheism is the view that the law of non- Edward N. Zalta contradiction is false. While there are plenty of philosophers who accept, Center for the Study of Language and Information and work within, paraconsistent logic, only a few count themselves as Stanford University dialetheists. [email protected] I take paraconsistent logic and dialetheism seriously, and think that they offer a philosophically worthy approach to these puzzling groups of sentences. The logical investigation of paraconsistent logic is certainly An important philosophical puzzle arises whenever we find a group interesting and justified. We should endeavor to know what are the of philosophically interesting sentences which individually appear to be metatheoretical features of this logic. Dialetheism also deserves careful true but jointly imply a contradiction. It is traditional to suppose that study. Truth value gluts may be no worse than truth value gaps,1 and it since the sentences in the group are jointly inconsistent, we cannot accept is always good to investigate whether, or why, philosophers just take it them all. This refusal to accept all the sentences in the group is not just on faith that no contradictions are true.
    [Show full text]
  • Paradox and Foundation Zach Weber Submitted in Total Fulfilment of The
    Paradox and Foundation Zach Weber Submitted in total fulfilment of the requirements of the degree of Doctor of Philosophy May 2009 School of Philosophy, Anthropology and Social Inquiry The University of Melbourne This is to certify that - the thesis comprises only my original work towards the PhD, - due acknowledgement has been made in the text to all other material used, - the thesis is less than 100,000 words in length. Preface Dialethic paraconsistency is an approach to formal and philosophical theories in which some but not all contradictions are true. Advancing that program, this thesis is about paradoxes and the foundations of mathematics, and is divided accordingly into two main parts. The first part concerns the history and philosophy of set theory from Cantor through the independence proofs, focusing on the set concept. A set is any col- lection of objects that is itself an object, with identity completely determined by membership. The set concept is called naive because it is inconsistent. I argue that the set concept is inherently and rightly paradoxical, because sets are both intensional and extensional objects: Sets are predicates in extension. All consistent characterizations of sets are either not explanatory or not coherent. To understand sets, we need to reason about them with an appropriate logic; paraconsistent naive set theory is situated as a continuation of the original foundational project. The second part produces a set theory deduced from an unrestricted compre- hension principle using the weak relevant logic DLQ, dialethic logic with quantifiers. I discuss some of the problems involved with embedding in DLQ, especially related to identity and substitution.
    [Show full text]